# Second Derivatives and Beyond Examples

#### Third Derivatives and Beyond

We're learning calculus, so, by definition, we're in the prime of our lives. Things are great. Things are awesome. We can take third derivatives, fourth derivatives, and so on. Writing f " f...

#### Critical Points

We say x = c is a critical point of the function f if f (c) exists and f '(c) = 0 or is undefined. It's generally a peak or valley in the curve. It's where the slopes becomes interestin...

#### Points of Inflection

A point of inflection or inflection point, abbreviated IP, is an x-value at which the concavity of the function changes. In other words, an IP is an x-value where the sign of the second derivativ...

#### First Derivative Test

Sample ProblemBelow is a graph of a function f with a minimum at x = x0. Determine the sign of the derivative f ' at each labelled x-value.Below is a graph of a function f with a maximum at...

#### Second Derivative Test

Sample ProblemsAssume f is defined and twice differentiable on the whole real line. Around a minimum of the function f, is f concave up or concave down? Assume f is defined and twice d...

#### Local vs. Global Points

Sometimes we take vacations. Sometimes we take stay-cations. Second derivatives can be used to determine if the function will be traveling somewhere extreme or if it will travel somewhere more su...

#### Finding Points

There are three types of points to find on the function, and the great thing is that you already know how to find all of them. Intercepts: To find the y-intercept of a function, plug x = 0 into t...

#### Finding Shapes

We talked earlier about how we can tell from the derivative whether the original function is increasing or decreasing. We also talked about concavity, and how if we toss the second derivative int...

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